1 Introduction
Tubular structures are ubiquitous throughout the human body, with notable examples including blood vessels, pancreatic duct and urinary tract. They occur in specific environments at the boundary of liquids, solids or air and surrounding tissues, and play a prominent role in sustaining physiological functions of the human body.
In this paper, we investigate automatic tubular organ/tissue segmentation from CT scans, which is important for the characterization of various diseases [18]. For example, pancreatic duct dilatation or abrupt pancreatic duct caliber change signifies high risk for pancreatic ductal adenocarcinoma (PDAC), which is the third most common cause of cancer death in the US [11]. Another example is that obstructed vessels lead to coronary heart disease, which is the leading cause of death in the US [27].
Segmenting tubular organs/tissues from CT scans is a popular but challenging problem. Existing methods addressing this problem can be roughly categorized into two groups: (1) Geometrybased methods, which build deformable shape models to fit tubular structures by exploiting their geometrical properties [42, 44, 3, 25], e.g., a tubular structure can be well represented by its skeleton, aka symmetry axis or medial axis
, and it has a cylindrical surface. But, due to the lack of powerful learning models, these methods cannot deal with poor contrast, noise and complicated background. (2) Learningbased methods, which learn a perpixel classification model to detect tubular structures. The performance of this type of methods is largely boosted by deep learning, especially fully convolutional networks (FCN)
[23, 47]. FCN and its variants have become outofthebox models for tubular organ/tissue segmentation and achieve stateoftheart results [24, 46]. However, these networks simply try to learn a class label per voxel, which inevitably ignores the geometric arrangement of the voxels in a tubular structure, and consequently can not guarantee that the obtained segmentation has the right shape.Since a tubular structure can be well represented by its skeleton and the crosssectional radius of each skeleton point, as shown in Fig. 1, these intrinsic geometric characteristics should be taken into account to serve as a valuable prior. To this end, a straightforward strategy is to first train a learning model, e.g., a deep network, to directly predict whether each voxel is on the skeleton of the tubular structure or not as well as the crosssectional radius of each skeleton point, and then reconstruct the segmentation of the tubular structure from its skeleton and radii [34]. However, such a strategy has severe limitations: (1) The groundtruth skeletons used for training are not easily obtained. Although they can be approximately computed from the groundtruth segmentation mask by 3D skeletonization methods, skeleton extraction from 3D mesh representation itself is a hard and unsolved problem [5]
. Without reliable skeleton groundtruths, the performance of tubular structure segmentation cannot be guaranteed. (2) It is hard for the classifier to distinguish voxels on the skeleton itself from those immediately next to it, as they have similar features but different labels.
To tackle the obstacles mentioned above, we propose to perform tubular structure segmentation by training a multitask deep network to predict not only a segmentation mask for a tubular structure, but also a distance map, consisting of the distance transform value from each tubular structure voxel to the tubular structure surface, rather than a single skeleton/nonskeleton label. Distance transform [28] is a classical image processing operator to produce a distance map with the same size of the input image, each value in which is the distance from each foreground pixel/voxel to the foreground boundary. Distance transform is also known as the basis of one type of skeletonization algorithms [17], i.e., the ridge of the distance map is the skeleton. Thus, the predicted distance map encodes the geometric characteristics of the tubular structure. This motivated us to design a geometryaware approach to refine the output segmentation mask by leveraging the shape prior reconstructed from the distance map. Essentially, our approach performs tubular structure segmentation by an implicit skeletonizationreconstruction procedure with no requirements for skeleton groundtruths. We stress that the distance transform brings two benefits for our approach: (1) Distance transform values are defined on each voxel inside a tubular structure, which eliminates the problem of the discontinuity between the skeleton and its surrounding voxels; (2) distance transform values on the skeleton (the ridge of the distance map) are exactly the crosssectional radii (scales) of the tubular structure, which is an important geometrical measurement. To make the distance transform value prediction more precise, we additionally propose a distance loss term used for network training, which indicates a penalty when predicted distance transform value is far away from its groundtruth.
We term our method Deep Distance Transform (DDT), as it naturally combines intuitions from the classical distance transform for skeletonization and modern deep segmentation networks. We emphasize that DDT has two advantages over vanilla segmentation networks: (1) It guides tubular structure segmentation by taking the geometric property of tubular structures into account. This reduces the difficulty to segment tubular structures from complex surrounding structures and ensures that the segmentation results have a proper shape prototype; (2) It predicts the crosssectional scales of a tubular structure as byproducts, which are important for the further study of the tubular structure, such as clinical diagnosis and virtual endoscopy [7].
We verify DDT on six datasets, including five datasets for segmentation task, and one dataset for clinical diagnosis. For segmentation task, the performance of our DDT exceeds all backbone networks by a large margin, with even over 13% improvement in terms of DiceSørensen coefficient for pancreatic duct segmentation on the famous 3DUnet [12]. The ablation study further shows the effectiveness of each proposed module in DDT. The experiment for clinical diagnosis leverages dilated pancreatic duct as cue for finding missing PDAC tumors by original deep networks, which verifies the potential of our DDT for early diagnosis of pancreatic cancer.
2 Related Work
2.1 Tubular Structure Segmentation
2.1.1 Geometrybased Methods
Various methods have been proposed to improve the performance of tubular structure segmentation by considering the geometric characteristics, and a nonexhaustive overview is given here. (1) Contourbased methods extracted the segmentation mask of a tubular structure by means of approximating its shape in the crosssectional domain [1, 10]. (2) Minimal path approaches conducted tubular structure tracking and were usually interactive. They captured the global minimum curve (energy weighted by the image potential) between two points given by the user [9]. (3) Modelbased tracking methods required to refine a tubular structure model, which most of the time adopted a 3D cylinder with elliptical or circular section. At each tracking step, they calculated the new model position by finding the best model match among all possible new model positions [8]
. (4) Centerline based methods found the centerline and estimated the radius of linear structures. For example, multiscale centerline detection method proposed in
[34] adopted the idea of distance transform, and reformulated centerline detection and radius estimation in terms of a regression problem in 2D. Our work fully leverages the geometric information of a tubular structure, proposing a distance transform algorithm to implicitly learn the skeleton and crosssectional radius, and the final segmentation mask is reconstructed by adopting the shape prior of the tubular structure.2.1.2 Learningbased Method
Learningbased method for tubular structure segmentation infers a rule from labeled training pairs, one for each pixel. Traditional methods such as 2D Gabor wavelet and classifier combination [35], ridgebased segmentation [36], and random decision forest based method [2] achieved considerable progress. In the past years, various 2D and 3D deep segmentation networks have become very popular. Some multiorgan segmentation methods [40, 29] were proposed to segment multiple organs simultaneously, including tubular organs. DeepVessel [16] put a fourstage HEDlike CNN and conditional random field into an integrated deep network to segment retinal vessel. KidNet [37], inspired from 3DUnet [12], was a twophase 3D network for kidney vessels segmentation. ResDSN [48, 49] and 3DUnet [12] were used in Hyperpairing network [46] to segment tissues in pancreas including duct by combining information from dualphase imaging. Besides, 3DHED and its variant were applied for vascular boundary detection [24]. Other scenarios such as using synthetic data to improve endotracheal tube segmentation [15]. Crossmodality domain adaptation framework with adversarial learning which dealt with the domain shift in segmenting biomedical images including ascending aorta was also proposed [13].
More powerful deep network architecture can produce better segmentation results. But how to leverage geometric information while employing the power of deep network is a more intriguing problem, especially for tubular structure. Our work aims at designing an integrated framework which mines traditional distance transform and model deep networks for such cylinderlike shape structure, which is not studied in prior research.
2.2 Learningbased Skeleton Extraction
Learningbased skeleton extraction from natural images has been widely studied in recent decades [38, 31, 34, 22, 21] and achieved promising progress with the help of deep learning [32, 20, 45, 39]. Shen et al. [32] showed that multitask learning, i.e., jointly learning skeleton pixel classification and skeleton scale regression, was important to obtain accurate predicted scales, and it was useful for skeletonbased object segmentation.
However, these methods cannot be directly applied to tubular structure segmentation, since they require the skeleton groundtruth, which is not easy to obtain from a 3D mask due to the commonly existed annotation errors for medical images [41].
3 Methodology
We first define a 3D volume of size as a function on the coordinate set , i.e., where the value on position is defined as . , , represent for the integer set ranging from to , , respectively, so that the Cartesian product of them can form the coordinate set. Given a 3D CT scan , the goal of tubular structure segmentation is to predict the label of all voxels in the CT scan, where denotes the predicted label for each voxel at position , i.e., if the voxel at is predicted as a tubular structure voxel, then , otherwise . We also use to denote the voxel at position in the remaining of the paper for convenience sake. Fig. 2 illustrates our tubular structure segmentation network, i.e., DDT.
3.1 Distance Transform for Tubular Structure
In this section, we discuss how to perform distance transform for tubular structure voxels. Given the groundtruth label map of the CT scan in the training phase, let be the set of voxels on the tubular structure surface, which can be defined by
(1) 
where denotes the 6neighbour voxels of . Then, by performing distance transform on the CT scan , the distance map is computed by
(2) 
Note that, for each tubular structure voxel , the distance transform assigns it a distance transform value which is the nearest distance from to the tubular structure surface . Here we use Euclidean distance, as skeletons from Euclidean distance maps are robust to rotations [4].
We further quantize each into one of bins by rounding to the nearest integer, which converts the continuous distance map to a discrete quantized distance map , where
. We do this quantization, because training a deep network directly for regression is relatively unstable, since outliers,
i.e., the commonly existed annotation errors for medical images [41], cause a large error term, which makes it difficult for the network to converge and leads to unstable predictions [30]. Based on quantization, we rephrase the distance prediction problem as a classification problem, i.e., to determine the corresponding bin for each quantized distance. We term the bins of the quantized distances as scale classes. We use the term scale since the distance transform values at the skeleton voxels of a tubular structure are its crosssectional scales.3.2 Network Training for Deep Distance Transform
Given a 3D CT scan and its groundtruth label map , we can compute its scale class map (quantized distance map) according to the method given in Sec. 3.1. In this section, we describe how to train a deep network for tubular structure segmentation by targeting on both and .
As shown in Fig. 2, our DDT model has two head branches. The first one is targeting on the groundtruth label map
, which performs pervoxel classification for semantic segmentation with a weighted crossentropy loss function
:(3) 
where is the parameters of the network backbone, is the parameters of this head branch and
is the probability that
is a tubular structure voxel as predicted by this head branch. and are loss weights for tubular structure and background classes respectively.The second head branch is predicting on the scale class map , which performs scale prediction for tubular structure voxels (i.e., ). We introduce a new distance loss function to learn this head branch:
(4) 
where is the parameters of the network backbone, is the parameters of the second head branch, is an indication function, is a tradeoff parameter which balances the two loss terms (we simply set in our implementation), is the probability that the scale of belongs to th scale class and is a normalized weight defined by . Note that, the first term of Eq. 3.2 is the standard softmax loss which penalizes the classification error for each scale class equally. The second term of Eq. 3.2 is termed as distance loss term, which penalizes the difference between each predicted scale class (i.e., ) and its groundtruth scale class , where the penalty is controlled by . Finally, the loss function for our segmentation network is and the optimal network parameters are obtained by .
3.3 Geometryaware Refinement
Given a 3D CT scan in the testing phase, for each voxel , our tubular structure segmentation network, DDT, outputs two probabilities, , which is the probability that is a tubular structure voxel and , which is the probability that the scale of belongs to th scale class. For notational simplicity, we use and to denote and , respectively, in the rest of the paper. provides pervoxel tubular structure segmentation, and encodes the geometric characteristics of the tubular structure. We introduce a geometryaware refinement approach to obtain the final segmentation result by refining according to . This approach is shown in Fig. 2 and is processed as follows:

[label=.]

Pseudo skeleton generation. The probability map is thinned by thresholding it to generate a binary pseudo skeleton map for the tubular structure. If , ; otherwise, , and is the threshold.

Shape reconstruction. For each voxel , its predicted scale is given by . It is known that a shape can be reconstructed from its skeleton by enveloping the maximal balls centered at each skeleton point, e.g., we can obtain the (binary) reconstructed shape of the tubular structure from by if , where is a ball centered at with radius ; otherwise . However, the predicted scale is quantized, which leads to an nonsmooth surface. Therefore, we fit a Gaussian kernel to soften each ball and obtain a soft reconstructed shape instead:
(5) where
denotes the density function of a multivariate normal distribution,
is the mean andis the covariance matrix. According to the 3sigma rule, we set
, whereis an identity matrix. We notice that the peak of
becomes smaller if is larger. To normalize the peak of each normal distribution, we introduce a normalization factor . 
Segmentation refinement. We use the soft reconstructed shape to refine the segmentation probability , which results in a refined segmentation map :
(6) The final segmentation mask is obtained by thresholding , i.e., if , , otherwise, , where and are the value of voxel at position of and , respectively.
4 Experiments
In this section, we conduct the following experiments: we first evaluate our approach on five segmentation datasets, including (1) the dataset used in [46], (2) three tubular structure datasets created by radiologists in our team, and (3) hepatic vessels dataset in Medical Segmentation Decathlon (MSD) challenge [33]. Then, as we mentioned in Sec. 1, our DDT predicts crosssectional scales as byproducts, which are important for applications such as clinical diagnosis. We show that the crosssectional scale is an important measurement for predicting the dilation degree of a pancreatic duct, which can help find the PDAC tumors missed in [49], without increasing the false positives.
4.1 Tubular Structure Segmentation
4.1.1 Implementation Details and Evaluation Metric
Our implementation is based on PyTorch. For data preprocessing, followed by
[46], we truncate the raw intensity values within the range of HU and normalize each CT scan into zero mean and unit variance. Data augmentation (i.e.,translation, rotation and flipping) is conducted in all the methods, leading to an augmentation factor of 24. During training, we randomly sample patches of a specified size (i.e., 64) due to memory issue. We use exponential learning rate decay with . During testing, we employ the sliding window strategy to obtain the final predictions. The groundtruth distance map for each tubular structure is computed by finding the euclidean distance of each foreground voxel to its nearest boundary voxels. The segmentation accuracy is measured by the wellknown DiceSørensen coefficient (DSC) in the rest of the paper, unless otherwise specified.4.1.2 The PDAC Segmentation Dataset [46]
We first study the PDAC segmentation dataset [46] which has 239 patients with pathologically proven PDAC. All CT scans are contrast enhanced images and our experiments are conducted on only portal venous phase. We follow the same setting and the same crossvalidation as reported in [46]. DSCs for three structures were reported in [46]
: abnormal pancreas, PDAC mass and pancreatic duct. We only show the average and standard deviation over all cases for pancreatic duct, which is a tubular structure.
Results and Discussions.
To evaluate the performance of the proposed DDT framework, we compare it with a pervoxel classification method [46], termed as SegBaseline in Table 1. It can be seen that our approach outperforms the baseline reported in [46] by a large margin. It is also worth mentioning that although our DDT is only tested on venous phase, the performance is comparable with the hyperparing network [46] (i.e., Multiphase HPN), which integrates multiphase information (i.e., arterial phase and venous phase). For 3DUNet, our DDT even outperforms the multiphase method by more than 13% in terms of DSC.
Methods  Phase  Backbone Networks  

3DUNet  ResDSN  
SegBaseline [46]  V  40.25 27.89  49.81 26.23 
Multiphase HPN [46]  A+V  44.93 24.88  56.77 23.33 
DDT (Ours)  V  58.20 23.39  55.97 24.76 
Ablation Study.
We conduct ablation experiments on the PDAC segmentation dataset, using ResDSN as the backbone. These variants of our methods are considered:

[leftmargin=*]

SegfromSkel: This is the straightforward strategy mentioned in Sec. 1 for skeletonbased tubular structure segmentation, i.e., segmenting by reconstructing from the predicted skeleton. The groundtruth skeleton is obtained by the mesh contraction algorithm [5], and the scale of each skeleton point is defined as its shortest distance to the duct surface. We use the same method in Sec. 3 to instantiate this strategy, but the learning target is the skeleton instead of the duct mask.

DDT , w/o GAR: DDT without distance loss term ( in Eq. 3.2), and without geometryaware refinement.

DDT , w/ GAR: DDT without distance loss term, and with geometryaware refinement.

DDT , w/o GAR: DDT with distance loss term, and without geometryaware refinement.

DDT , w/ GAR: DDT with distance loss term, and with geometryaware refinement.
The results of the ablation experiments are summarized in Table 2. Then, we aim at discussing parameters in the geometryaware refinement component. In our implementation, we set and in Sec. 3.3. Now we vary each of them and fix the other one to the default value to see how the performance changes. As shown in Fig. 3(a), setting a larger leads to better performance. This phenomenon further verifies the advantage of leveraging scale class map to refine the pervoxel segmentation results, i.e., a thinner pseudo skeleton combined with a scale class map can better represent a tubular structure. Fig. 3(b) shows that the performance is not sensitive within the range of .
Method  Average DSC (%) 

SegBaseline [46]  49.81 
SegfromSkel  51.88 
DDT , w/o GAR  52.73 
DDT , w/ GAR  54.70 
DDT , w/o GAR  53.69 
DDT , w/ GAR  55.97 
Backbone  Methods  Aorta  Veins  Pancreatic duct  

Average  Mean Surface  Average  Mean Surface  Average  Mean surface  
DSC  Distance  DSC  Distance  DSC  Distance  
3DHED [24]  SegBaseline  90.85  1.15  73.57  5.13  46.43  7.06 
DDT  92.94  0.82  76.20  3.78  54.43  4.91  
3DUNet [12]  SegBaseline  92.01  0.94  71.57  4.46  56.63  3.64 
DDT  93.30  0.61  75.59  4.07  62.31  3.56  
ResDSN [48]  SegBaseline  89.89  1.12  71.10  6.25  55.91  4.24 
DDT  92.57  1.10  76.60  5.03  59.29  4.19 
4.1.3 Tubular Structure Datasets
We then evaluate our algorithm on multiple tubular structures datasets. Radiologists in our team collected 229 abdominal CT scans of normal cases with aorta annotation, 204 normal cases with veins annotation, and 494 abdominal CT scans of biopsyproven PDAC cases with pancreatic duct annotation. All these three datasets are under IRB approved protocol.
We conduct experiments by comparing our DDT with SegBaseline on three backbone networks: 3DHED [24], 3DUNet [12] and ResDSN [48, 49]. The results in terms of DSC and mean surface distance (in mm) are reported in Table 3. SegBaseline methods on all backbone networks are significantly lower than our approach. In particular, for 3DHED, DDT outperforms SegBaseline by 8%, making its strong ability in segmenting small tubular structures like pancreatic duct in medical images. The results are obtained by crossvalidation. We also illustrate segmentation results of aorta and veins in Fig. 4 for qualitative comparison. We can see that compared with SegBaseline, DDT captures geometry information, which is more robust to the noise and complicated background.
4.1.4 Hepatic Vessels Dataset in MSD Challenge
We also test our DDT on a public hepatic vessels dataset in MSD challenge [33]. There are two targets in hepatic vessels dataset: vessels and tumor. As our goal is to segment tubular structure, we aim at vessel segmentation. Although this challenge is over, it is still open for submissions. We train our DDT on 303 training cases, and submit vessel predictions of the testing cases to the challenge.
4.2 Finding PDAC Tumor by Dilated Duct
Methods  Average DSC () 

DDT (Ours)  63.43 
nnUNet [19]  63.00 
UMCT [43]  63.00 
K.A.V.athlon  62.00 
LS Wang’s Group  55.00 
MIMI  60.00 
MPUnet [26]  59.00 
Background.
PDAC is one of the most deadly disease, whose survival is dismal as more than of patients have evidence of metastatic disease at the time of diagnosis. As mentioned in Sec. 1, dilated duct is a vital cue for the presence of a PDAC tumor. The reason lies in that in most cases, the tumor blocks the duct and causes it to dilate, as shown in Fig. 5. Experienced radiologists usually trace the duct from the pancreas tail onward to see if there exists a truncated duct. If they see the predicted duct pattern as illustrated in Fig. 5, they will be alarmed and treat it as a suspicious PDAC case. For computeraided diagnosis, given a mixture of normal and abnormal CT scans (PDAC cases), if some voxels are segmented as a tumor by a stateoftheart deep network, we can provide radiologists with tumor locations [49]. But, as reported [49], even a stateoftheart deep network failed to detect 8 PDACs out of 136 abnormal cases. As emphasized in [49], for clinical purposes, we shall guarantee a high sensitivity with a reasonable specificity. Then how can we use dilated duct as a cue to help find the PDAC tumor in an abnormal case even if it does NOT have any PDAC tumor prediction by directly applying deep networks?
Clinical Workflow.
The flowchart of our strategy is illustrated in Fig. 6. We apply our DDT on the cases which do not have tumor prediction by [49]. Then the predictions of DDT are processed as follows:

[leftmargin=*]

Find cases with predicted dilated duct. Let’s assume a case has predicted duct voxels. If , then we regard this case as negative. If , let’s denote the predicted associated scales (radii) are . If , i.e., the largest crosssectional scale is larger than , we regard this is a dilated duct, and a tumor may present on its head location. Otherwise, we treat this case as negative. We set , since the radius of a dilated duct should be larger than 1.5 mm [14], and the voxel spatial resolution of the dataset [49] is around 0.5 mm.

Extract candidate tumor regions by the location of dilated duct. We use geodesic distance to find the extreme points of the duct [6]. Then we crop a set of square regions of size centered on the extreme points not lying on the tail of the pancreas, since a tumor presenting on the tail of the pancreas will not block a duct. This set of square regions are candidate tumor regions.

Verify candidate tumor regions. As candidate tumor regions may come from both normal and abnormal cases. We should verify whether the candidate region is a real tumor region. From the training set, we randomly crop regions of size around PDAC tumor region as positive training data, and randomly crop regions of size from normal pancreas as negative training data. Then we train a ResDSN [49] to verify these candidate tumor regions. We follow the same criterion used in [49] to compute sensitivity and specificity.
Experiment Settings.
Analysis.
We compare our results with those of [49] in Table 5. In our experiment, 4 out of 8 abnormal cases and 3 out of 197 normal cases have predicted dilated duct by step 1. An example is shown in Fig. 7(a). The tubular structure residing inside the groundtruth pancreas, right behind the groundtruth tumor is our predicted dilated duct. This leads to overall 18 3D candidate tumor regions by step 2, shown as the yellow dashed box in Fig. 7(b) visualized in 2D. In step 3, we can successfully find all tumor regions in abnormal cases, and discard nontumor regions in normal cases. As shown in Fig. 7(c), our algorithm can find the right tumor, which overlaps with the tumor annotation in Fig. 7(d).
It should be emphasized that dilated duct helps us to narrow the searching space of the tumor, so that we are able to focus on a finer region. Though we train a same network used in [49], half of the missing tumors in [49] can be found. In this way, we are imitating how radiologists detect PDAC, i.e., they can find visible PDAC tumors easily, but for difficult ones, they will seek help from dilated ducts.
Methods  Misses  Sensitivity  Specificity 

Zhu et al. [49]  8/136  94.1%  98.5% 
Zhu et al. [49] + Ours  4/136  97.1%  98.5% 
5 Conclusions
In this paper, we present Deep Distance Transform (DDT) for accurate tubular structure segmentation, which combines intuitions from the classical distance transform for skeletonization and modern deep segmentation networks. DDT guides tubular structure segmentation by taking the geometric property of the tubular structure into account, which not only leads to a better segmentation result, but also provides the crosssectional scales, i.e., a geometric measure for the thickness of tubular structures. We evaluated our approach on six datasets including four tubular structures: pancreatic duct, aorta, veins and vessels. Experiment shows the superiority of the proposed DDT for tubular structure segmentation and clinical application.
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